The table below gives values of the functions $f$ and $g$ at different values of $x$. What is $f(g(1))$ ?

$$\begin{array}{c|c|c|c} x & 1 & 2 & 3 \\ \hline f(x) & 3 & -6 & 2 \\ g(x) & 2 & 7 & 1 \end{array}$$

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From the table, $g(1)=2$.

$$f(g(1)) = f(2)$$

From the table, $f(2)=\boxed{-6}$.